Operator Algebras and Their Modules is intended for a specific audience: professional mathematicians whose specialty is some form of functional analysis. In order to appreciate this book, the reader must already possess a broad knowledge of Hilbert spaces, Banach spaces, Banach algebras, and C*-algebras. (These topics are discussed somewhat in an appendix at the end of the book, but in a highly abbreviated fashion.) For a reader with the necessary background and predisposition, this book could prove to be quite an asset.

The authors waste no time in getting to the study of serious mathematics. By the end of the first page, it is clear that the book will demand a substantial commitment on the part of the reader. The introductory chapter on operator spaces, in fact, could serve as the basis for a graduate-level course. The next two chapters provide similar introductions to operator algebras and operator modules. The book continues with more specialized topics, such as tensor products of operator algebras, self-adjointness criteria, and C*-modules and operator spaces. Each chapter concludes with a lengthy section of notes and historical remarks; these are particularly interesting in light of the authors’ own influence on the evolution of these concepts.

The exposition throughout the book is thorough and comprehensive. The authors have clearly invested a great deal of time and effort into this project, and the final result reflects this fact. The quality of the prose is excellent. The text is organized in a logical, coherent, and comprehensible manner. The proofs have been refined almost to the point of maximal efficiency. There is virtually no superfluous material anywhere in the text; indeed, the authors have made the most of the 358 pages allotted them.

This book, while only appropriate for a fairly small audience, is a significant contribution to the literature of the discipline. Anyone who masters this book will, almost of necessity, become an expert in this area of mathematics.

Christopher Hammond is Assistant Professor of Mathematics at Connecticut College.